Question

1. using factoring, find the solution for:

\\(\frac{4x^2 + 83x}{4} = -15\\)

write all solutions on the same line, separated by commas.

enter your next step here

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the equation \(\frac{4x^{2}+83x}{4}=-15\) by 4 to get rid of the denominator.
\(4\times\frac{4x^{2}+83x}{4}=4\times(-15)\)
Simplifying, we have \(4x^{2}+83x = - 60\).

Step2: Rearrange into standard quadratic form

Add 60 to both sides to set the equation to zero: \(4x^{2}+83x + 60=0\).

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \(4\times60 = 240\) and add up to 83. The numbers are 80 and 3.
Rewrite the middle term using these numbers: \(4x^{2}+80x+3x + 60 = 0\).
Group the terms: \((4x^{2}+80x)+(3x + 60)=0\).
Factor out the common factors from each group: \(4x(x + 20)+3(x + 20)=0\).
Now, factor out \((x + 20)\): \((4x + 3)(x + 20)=0\).

Step4: Solve for x

Set each factor equal to zero:

• \(4x+3 = 0\) gives \(4x=-3\), so \(x=-\frac{3}{4}\).
• \(x + 20=0\) gives \(x=-20\).

Answer:

\(-20,-\frac{3}{4}\)